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Data Mining and Machine Learning with EM

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Data Mining and Machine Learning are Ubiquitous! Netflix Amazon Wal-Mart Algorithmic Trading/High Frequency Trading Banks (Segmint) Google/Yahoo/Microsoft/IBM CRM/Consumer Behavior Profiling Consumer Review Mobile Ads Social Network (Facebook/Twitter/Google+) Voting Behaviors …

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Data Mining Non-trivial extraction of implicit, previously unknown and potentially useful information from data Exploration & analysis, by automatic or semi-automatic means, of large quantities of data in order to discover meaningful patterns

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Data Mining Tasks Prediction Methods – Use some variables to predict unknown or future values of other variables. Description Methods – Find human-interpretable patterns that describe the data. From [Fayyad, et.al.] Advances in Knowledge Discovery and Data Mining, 1996

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Data Mining Tasks... Classification [Predictive] Clustering [Descriptive] Association Rule Discovery [Descriptive] Sequential Pattern Discovery [Descriptive] Regression [Predictive] Deviation Detection [Predictive]

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Association Rule Discovery: Definition Given a set of records each of which contain some number of items from a given collection; – Produce dependency rules which will predict occurrence of an item based on occurrences of other items. Rules Discovered: {Milk} --> {Coke} {Diaper, Milk} --> {Beer} Rules Discovered: {Milk} --> {Coke} {Diaper, Milk} --> {Beer}

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Association Rule Discovery: Application 1 Marketing and Sales Promotion: – Let the rule discovered be {Bagels, … } --> {Potato Chips} – Potato Chips as consequent => Can be used to determine what should be done to boost its sales. – Bagels in the antecedent => C an be used to see which products would be affected if the store discontinues selling bagels. – Bagels in antecedent and Potato chips in consequent => Can be used to see what products should be sold with Bagels to promote sale of Potato chips!

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9 Definition: Frequent Itemset Itemset – A collection of one or more items Example: {Milk, Bread, Diaper} – k-itemset An itemset that contains k items Support count ( ) – Frequency of occurrence of an itemset – E.g. ({Milk, Bread,Diaper}) = 2 Support – Fraction of transactions that contain an itemset – E.g. s({Milk, Bread, Diaper}) = 2/5 Frequent Itemset – An itemset whose support is greater than or equal to a minsup threshold

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Frequent Itemsets Mining TIDTransactions 100{ A, B, E } 200{ B, D } 300{ A, B, E } 400{ A, C } 500{ B, C } 600{ A, C } 700{ A, B } 800{ A, B, C, E } 900{ A, B, C } 1000{ A, C, E } Minimum support level 50% – {A},{B},{C},{A,B}, {A,C}

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11 Frequent Itemset Generation Given d items, there are 2 d possible candidate itemsets

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12 Frequent Itemset Generation Brute-force approach: – Each itemset in the lattice is a candidate frequent itemset – Count the support of each candidate by scanning the database – Match each transaction against every candidate – Complexity ~ O(NMw) => Expensive since M = 2 d !!!

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13 Reducing Number of Candidates Apriori principle: – If an itemset is frequent, then all of its subsets must also be frequent Apriori principle holds due to the following property of the support measure: – Support of an itemset never exceeds the support of its subsets – This is known as the anti-monotone property of support

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14 Illustrating Apriori Principle Found to be Infrequent Pruned supersets

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Apriori R. Agrawal and R. Srikant. Fast algorithms for mining association rules. VLDB, , 1994Fast algorithms for mining association rules

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What is Cluster Analysis? Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups Inter-cluster distances are maximized Intra-cluster distances are minimized

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Applications of Cluster Analysis Understanding – Group related documents for browsing, group genes and proteins that have similar functionality, or group stocks with similar price fluctuations Summarization – Reduce the size of large data sets Clustering precipitation in Australia

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Notion of a Cluster can be Ambiguous How many clusters? Four ClustersTwo Clusters Six Clusters

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Types of Clusterings A clustering is a set of clusters Important distinction between hierarchical and partitional sets of clusters Partitional Clustering – A division data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subset Hierarchical clustering – A set of nested clusters organized as a hierarchical tree

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Partitional Clustering Original Points A Partitional Clustering

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Hierarchical Clustering Traditional Hierarchical Clustering Non-traditional Hierarchical ClusteringNon-traditional Dendrogram Traditional Dendrogram

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K-means Clustering Partitional clustering approach – Each cluster is associated with a centroid (center point) – Each point is assigned to the cluster with the closest centroid Number of clusters, K, must be specified The basic algorithm is very simple

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K-means Clustering – Details Initial centroids are often chosen randomly. – Clusters produced vary from one run to another. The centroid is (typically) the mean of the points in the cluster. Closeness is measured by Euclidean distance, cosine similarity, correlation, etc.

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K-means Clustering – Details K-means will converge for common similarity measures mentioned above. Most of the convergence happens in the first few iterations. – Often the stopping condition is changed to Until relatively few points change clusters Complexity is O( n * K * I * d ) – n = number of points, K = number of clusters, I = number of iterations, d = number of attributes

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K-Means Clustering 31

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How to MapReduce K-Means? Given K, assign the first K random points to be the initial cluster centers Assign subsequent points to the closest cluster using the supplied distance measure Compute the centroid of each cluster and iterate the previous step until the cluster centers converge within delta Run a final pass over the points to cluster them for output

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K-Means Map/Reduce Design Driver – Runs multiple iteration jobs using mapper+combiner+reducer – Runs final clustering job using only mapper Mapper – Configure: Single file containing encoded Clusters – Input: File split containing encoded Vectors – Output: Vectors keyed by nearest cluster Combiner – Input: Vectors keyed by nearest cluster – Output: Cluster centroid vectors keyed by cluster Reducer (singleton) – Input: Cluster centroid vectors – Output: Single file containing Vectors keyed by cluster

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Mapper - mapper has k centers in memory. Input Key-value pair (each input data point x). Find the index of the closest of the k centers (call it iClosest). Emit: (key,value) = (iClosest, x) Reducer(s) – Input (key,value) Key = index of center Value = iterator over input data points closest to ith center At each key value, run through the iterator and average all the Corresponding input data points. Emit: (index of center, new center)

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Improved Version: Calculate partial sums in mappers Mapper - mapper has k centers in memory. Running through one input data point at a time (call it x). Find the index of the closest of the k centers (call it iClosest). Accumulate sum of inputs segregated into K groups depending on which center is closest. Emit: (, partial sum) Or Emit(index, partial sum) Reducer – accumulate partial sums and Emit with index or without

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Issues and Limitations for K-means How to choose initial centers? How to choose K? How to handle Outliers? Clusters different in – Shape – Density – Size

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Two different K-means Clusterings Sub-optimal ClusteringOptimal Clustering Original Points

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Importance of Choosing Initial Centroids

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Importance of Choosing Initial Centroids …

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Solutions to Initial Centroids Problem Multiple runs – Helps, but probability is not on your side Sample and use hierarchical clustering to determine initial centroids Select more than k initial centroids and then select among these initial centroids – Select most widely separated Postprocessing Bisecting K-means – Not as susceptible to initialization issues

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EM-Algorithm

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What is MLE? Given – A sample X={X 1, …, X n } – A vector of parameters θ We define – Likelihood of the data: P(X | θ) – Log-likelihood of the data: L(θ)=log P(X|θ) Given X, find

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MLE (cont) Often we assume that X i s are independently identically distributed (i.i.d.) Depending on the form of p(x|θ), solving optimization problem can be easy or hard.

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An easy case Assuming – A coin has a probability p of being heads, 1-p of being tails. – Observation: We toss a coin N times, and the result is a set of Hs and Ts, and there are m Hs. What is the value of p based on MLE, given the observation?

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An easy case (cont) p= m/N

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EM: basic concepts

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Basic setting in EM X is a set of data points: observed data Θ is a parameter vector. EM is a method to find θ ML where Calculating P(X | θ) directly is hard. Calculating P(X,Y|θ) is much simpler, where Y is hidden data (or missing data).

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The basic EM strategy Z = (X, Y) – Z: complete data (augmented data) – X: observed data (incomplete data) – Y: hidden data (missing data)

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The log-likelihood function L is a function of θ, while holding X constant:

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The iterative approach for MLE In many cases, we cannot find the solution directly. An alternative is to find a sequence: s.t.

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Jensens inequality

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log is a concave function

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Maximizing the lower bound The Q function

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The Q-function Define the Q-function (a function of θ): – Y is a random vector. – X=(x 1, x 2, …, x n ) is a constant (vector). – Θ t is the current parameter estimate and is a constant (vector). – Θ is the normal variable (vector) that we wish to adjust. The Q-function is the expected value of the complete data log-likelihood P(X,Y|θ) with respect to Y given X and θ t.

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The inner loop of the EM algorithm E-step: calculate M-step: find

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L(θ) is non-decreasing at each iteration The EM algorithm will produce a sequence It can be proved that

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The inner loop of the Generalized EM algorithm (GEM) E-step: calculate M-step: find

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Recap of the EM algorithm

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Idea #1: find θ that maximizes the likelihood of training data

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Idea #2: find the θ t sequence No analytical solution iterative approach, find s.t.

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Idea #3: find θ t+1 that maximizes a tight lower bound of a tight lower bound

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Idea #4: find θ t+1 that maximizes the Q function Lower bound of The Q function

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The EM algorithm Start with initial estimate, θ 0 Repeat until convergence – E-step: calculate – M-step: find

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Important classes of EM problem Products of multinomial (PM) models Exponential families Gaussian mixture …

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Probabilistic Latent Semantic Analysis (PLSA) PLSA is a generative model for generating the co- occurrence of documents d D={d 1,…,d D } and terms w W={w 1,…,w W }, which associates latent variable z Z={z 1,…,z Z }. The generative processing is: w1w1 w1w1 w2w2 w2w2 wWwW wWwW … d1d1 d1d1 d2d2 d2d2 dDdD dDdD … z1z1 z2z2 zZzZ P(d)P(d) P(z|d)P(z|d)P(w|z)P(w|z)

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Model The generative process can be expressed by: Two independence assumptions: 1)Each pair (d,w) are assumed to be generated independently, corresponding to bag-of-words 2)Conditioned on z, words w are generated independently of the specific document d.

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Model Following the likelihood principle, we detemines P(z), P(d|z), and P(w|z) by maximization of the log- likelihood function co-occurrence times of d and w. Observed data Unobserved data P(d), P(z|d), and P(w|d)

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Maximum-likelihood Definition – We have a density function P(x| Θ ) that is govened by the set of parameters Θ, e.g., P might be a set of Gaussians and Θ could be the means and covariances – We also have a data set X={x 1,…,x N }, supposedly drawn from this distribution P, and assume these data vectors are i.i.d. with P. – Then the likehihood function is: – The likelihood is thought of as a function of the parameters Θwhere the data X is fixed. Our goal is to find the Θthat maximizes L. That is

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Jensens inequality

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Estimation-using EM difficult!!! Idea: start with a guess t, compute an easily computed lower-bound B( ; t ) to the function log P( |U) and maximize the bound instead By Jensens inequality:

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(1)Solve P(w|z) We introduce Lagrange multiplier λwith the constraint that w P(w|z)=1, and solve the following equation:

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(2)Solve P(d|z) We introduce Lagrange multiplier λwith the constraint that d P(d|z)=1, and get the following result:

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(3)Solve P(z) We introduce Lagrange multiplier λwith the constraint that z P(z)=1, and solve the following equation:

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(1)Solve P(z|d,w) We introduce Lagrange multiplier λwith the constraint that z P(z|d,w)=1, and solve the following equation:

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(4)Solve P(z|d,w) -2

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The final update Equations E-step: M-step:

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Coding Design Variables: double[][] p_dz_n // p(d|z), |D|*|Z| double[][] p_wz_n // p(w|z), |W|*|Z| double[] p_z_n // p(z), |Z| Running Processing: 1.Read dataset from file ArrayList doc; // all the docs DocWordPair – (word_id, word_frequency_in_doc) 2.Parameter Initialization Assign each elements of p_dz_n, p_wz_n and p_z_n with a random double value, satisfying d p_dz_n=1, d p_wz_n =1, and d p_z_n =1 3.Estimation (Iterative processing) 1.Update p_dz_n, p_wz_n and p_z_n 2.Calculate Log-likelihood function to see where ( |Log-likelihood – old_Log-likelihood| < threshold) 4.Output p_dz_n, p_wz_n and p_z_n

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Coding Design Update p_dz_n For each doc d{ For each word w included in d { denominator = 0; nominator = new double[Z]; For each topic z { nominator[z] = p_dz_n[d][z]* p_wz_n[w][z]* p_z_n[z] denominator +=nominator[z]; } // end for each topic z For each topic z { P_z_condition_d_w = nominator[j]/denominator; nominator_p_dz_n[d][z] += tf wd *P_z_condition_d_w; denominator_p_dz_n[z] += tf wd *P_z_condition_d_w; } // end for each topic z }// end for each word w included in d }// end for each doc d For each doc d { For each topic z{ p_dz_n_new[d][z] = nominator_p_dz_n[d][z]/ denominator_p_dz_n[z]; } // end for each topic z }// end for each doc d

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Coding Design Update p_wz_n For each doc d{ For each word w included in d { denominator = 0; nominator = new double[Z]; For each topic z { nominator[z] = p_dz_n[d][z]* p_wz_n[w][z]* p_z_n[z] denominator +=nominator[z]; } // end for each topic z For each topic z { P_z_condition_d_w = nominator[j]/denominator; nominator_p_wz_n[w][z] += tf wd *P_z_condition_d_w; denominator_p_wz_n[z] += tf wd *P_z_condition_d_w; } // end for each topic z }// end for each word w included in d }// end for each doc d For each w { For each topic z{ p_wz_n_new[w][z] = nominator_p_wz_n[w][z]/ denominator_p_wz_n[z]; } // end for each topic z }// end for each doc d

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Coding Design Update p_z_n For each doc d{ For each word w included in d { denominator = 0; nominator = new double[Z]; For each topic z { nominator[z] = p_dz_n[d][z]* p_wz_n[w][z]* p_z_n[z] denominator +=nominator[z]; } // end for each topic z For each topic z { P_z_condition_d_w = nominator[j]/denominator; nominator_p_z_n[z] += tf wd *P_z_condition_d_w; } // end for each topic z denominator_p_z_n[z] += tf wd ; }// end for each word w included in d }// end for each doc d For each topic z{ p_dz_n_new[d][j] = nominator_p_z_n[z]/ denominator_p_z_n; } // end for each topic z

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Apache Mahout Industrial Strength Machine Learning May 2008

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Current Situation Large volumes of data are now available Platforms now exist to run computations over large datasets (Hadoop, HBase) Sophisticated analytics are needed to turn data into information people can use Active research community and proprietary implementations of machine learning algorithms The world needs scalable implementations of ML under open license - ASF

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History of Mahout Summer 2007 – Developers needed scalable ML – Mailing list formed Community formed – Apache contributors – Academia & industry – Lots of initial interest Project formed under Apache Lucene – January 25, 2008

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Current Code Base Matrix & Vector library – Memory resident sparse & dense implementations Clustering – Canopy – K-Means – Mean Shift Collaborative Filtering – Taste Utilities – Distance Measures – Parameters

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Under Development Naïve Bayes Perceptron PLSI/EM Genetic Programming Dirichlet Process Clustering Clustering Examples Hama (Incubator) for very large arrays

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Appendix From Mahout Hands on, by Ted Dunning and Robin Anil, OSCON 2011, Portland

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Step 1 – Convert dataset into a Hadoop Sequence File ctions/reuters21578/reuters21578.tar.gz ctions/reuters21578/reuters21578.tar.gz Download (8.2 MB) and extract the SGML files. –$ mkdir -p mahout-work/reuters-sgm –$ cd mahout-work/reuters-sgm && tar xzf../reuters21578.tar.gz && cd.. && cd.. Extract content from SGML to text file –$ bin/mahout org.apache.lucene.benchmark.utils.ExtractReuter s mahout-work/reuters-sgm mahout-work/reuters- out

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Step 1 – Convert dataset into a Hadoop Sequence File Use seqdirectory tool to convert text file into a Hadoop Sequence File –$ bin/mahout seqdirectory \ -i mahout-work/reuters-out \ -o mahout-work/reuters-out-seqdir \ -c UTF-8 -chunk 5

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Hadoop Sequence File Sequence of Records, where each record is a pair – –… – Key and Value needs to be of class org.apache.hadoop.io.Text – Key = Record name or File name or unique identifier – Value = Content as UTF-8 encoded string TIP: Dump data from your database directly into Hadoop Sequence Files (see next slide)

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Writing to Sequence Files Configuration conf = new Configuration(); FileSystem fs = FileSystem.get(conf); Path path = new Path("testdata/part-00000"); SequenceFile.Writer writer = new SequenceFile.Writer( fs, conf, path, Text.class, Text.class); for (int i = 0; i < MAX_DOCS; i++) writer.append(new Text(documents(i).Id()), new Text(documents(i).Content())); } writer.close();

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Generate Vectors from Sequence Files Steps 1.Compute Dictionary 2.Assign integers for words 3.Compute feature weights 4.Create vector for each document using word-integer mapping and feature-weight Or Simply run $ bin/mahout seq2sparse

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Generate Vectors from Sequence Files $ bin/mahout seq2sparse \ -i mahout-work/reuters-out-seqdir/ \ -o mahout-work/reuters-out-seqdir-sparse-kmeans Important options – Ngrams – Lucene Analyzer for tokenizing – Feature Pruning Min support Max Document Frequency Min LLR (for ngrams) – Weighting Method TF v/s TFIDF lp-Norm Log normalize length

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Start K-Means clustering $ bin/mahout kmeans \ -i mahout-work/reuters-out-seqdir-sparse-kmeans/tfidf- vectors/ \ -c mahout-work/reuters-kmeans-clusters \ -o mahout-work/reuters-kmeans \ -dm org.apache.mahout.distance.CosineDistanceMeasure –cd 0.1 \ -x 10 -k 20 –ow Things to watch out for – Number of iterations – Convergence delta – Distance Measure – Creating assignments

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Inspect clusters $ bin/mahout clusterdump \ -s mahout-work/reuters-kmeans/clusters- 9 \ -d mahout-work/reuters-out-seqdir- sparse-kmeans/dictionary.file-0 \ -dt sequencefile -b 100 -n 20 Typical output :VL-21438{n=518 c=[0.56:0.019, 00:0.154, 00.03:0.018, 00.18:0.018, … Top Terms: iran => strike => iranian => union => said => workers => gulf => had => he =>

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